5. In the lecture, we have shown the following: Theorem: Suppose strategy s, (in a finite game) is strictly dominated by strat- egy ŝ; when opponents are restricted to using pure strategies. That is, Then strategy s, will never be in the support of a best response. That is, for all 0-i o; €BR;(G-;) = 0;(s})=0. (a) Use this result to find all the Nash equilibria (including mixed strategy ones) of the following game. player 2 L R T 10,5 4,1 player 1 M 6,1 8,8 B 5,0 0,10

5. In the lecture, we have shown the following: Theorem: Suppose strategy s, (in a finite game) is strictly dominated by strat- egy ŝ; when opponents are restricted to using pure strategies. That is, Then strategy s, will never be in the support of a best response. That is, for all 0-i o; €BR;(G-;) = 0;(s})=0. (a) Use this result to find all the Nash equilibria (including mixed strategy ones) of the following game. player 2 L R T 10,5 4,1 player 1 M 6,1 8,8 B 5,0 0,10

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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(d) Consider a simultaneous-move game between two firms choosing to sell their product at either £6, £7 or £8. The actions and payoffs are given in the matrix below. Firm 2's Prices £6 £7 £8 Firm 1's prices £6 4, 5 3, 5 2, 1 £7 0,4 2, 1 3,0 £8 -1, 1 4, 3 0, 2 What are the Nash equilibria of this game? Game theory is often used by firms competing under an oligopoly as a means of determining their best strategy. Why is game theory a useful tool and which characteristics of an oligopoly make it particularly useful for firms competing in this market structure? One outcome of an oligopoly is that firms may have an incentive to collude. Explain some of the conditions that make collusion more likely to occur and how game theory can explain why collusive agreements often break down.
2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.
Consider the following dynamic game: A (2,0) M 2 K (3,1) B 1 (1,4) A R (5,2) 2 K 1 C (3,1) (2,5) (1,0) (a) What is the strategy set for each player? How many strategies does each player have? (b) Find SPNE. (c) Is there a NE in which action C is a part of equilibrium strategy? (d) Give an example of some other NE that is not SPNE.
3.Students A and B decide whether to clean the dormitory. Regardless of whether the other party participates, the cleaning cost for each student to clean the dormitory is $ 8; and the benefit for each student from cleaning is $5 times the number of participants. (hint: if both of them clean the dormitory together, A and B benefit. If one cleans while the other does not, A and B benefit too) (a) Please use a game theory to briefly describe the above scenario, (that is to calculate the revenue for different strategies) (b) Draw the payoff matrix, find the Nash equilibrium.
Consider the following sequential game: 1 O D,L U.L U,R U D,R D (0,2) 2 L What is the subgame perfect Nash equilibrium of this game? R. (-1,-1) (1,1)
(a) Consider the following bimatrix of a normal form game. Show that the set of strategies that survive elimination of weakly dominated strategies depends on the order in which the weakly dominated strategies are eliminated. Player 2 L R 1,1 0,0 M| 1,1 0,0 | 2,1 Player 1 2,1 (b) Find an example of a game where the unique Nash equilibrium in pure strategies would not survive the eliminination of weakly dominated strategies. (c) Find an example of a game where both players' strategies are weakly dominated in the Nash equilibrium.
rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.
Which one of following statements is false? * Consider the following game. B D E F There are 7 subgames in this extensive-form game. There are 6 proper subgames in this extensive-form game. (BK, CE) is a Subgame Perfect Nash Equilibrium. O (BK, DE) is a Subgame Perfect Nash Equilibrium. G H I J K L M N 3,4 4,3 1,2 -5,7 6,5 2,8 9,1 3,6
2. Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A's payoff, and the second entry is Animal B's payoff: Animal A Hawk Dove (rows)/Animal B (columns) (-5,-5) (0,10) (10,0) (4,4) Hawk Dove An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. (a) Explain why there cannot be an equilibrium where all animals act as Doves. (b) Explore whether there are any Nash equilibria in pure strategies and explain which these are and why. (c) Derive a mixed strategy Nash equilibrium (MSNE). What is the proportion of Hawks and Doves? If the proportion of Hawks in the population of animals is greater than the mixed strategy equilibrium proportion you calculated, which strategy does better, Hawks of Doves? Explain your answer. (d) Draw the best response functions and show in the diagram all pure and mixed…
2. Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A's payoff, and the second entry is Animal B's payoff: Animal A Hawk Dove (rows)/Animal B (columns) (-5,-5) (0,10) (10,0) (4,4) Hawk Dove An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. (a) Explain why there cannot be an equilibrium where all animals act as Doves. (b) Explore whether there are any Nash equilibria in pure strategies and explain which these are and why. (c) Derive a mixed strategy Nash equilibrium (MSNE). What is the proportion of Hawks and Doves? If the proportion of Hawks in the population of animals is greater than the mixed strategy equilibrium proportion you calculated, which strategy does better, Hawks of Doves? Explain your answer.
a) Write out the extensive form of a game between a farmer (playing in the first round) and nature (playing a mixed strategy in the second round). Assume that the farmer can either pay a cash rent of $1500 for land (English system) or 1/2 of crop production to the landlord (sharecropping). Assume the farmer is planting corn and will produce 2 tons of corn. Assume that nature has a 50% chance of playing a strategy in which the price of corn is $3500/ton and a 50% chance of playing a strategy in which the price of corn is $500/ton. The farmer keeps any money left after paying cash rent and sells any corn left after paying the landlord in sharecropping. b)What is the most that a risk neutral farmer would be willing to pay for an accurate prediction of the price of corn in problem 1 before choosing whether to pay cash rent or sharecrop?
1. Consider the following N-player game. Each agent has a choice of strategy A or B. The state variable is x, the proportion of agents choosing strategy A. For every agent, the utility of choosing strategy A is U(A) = 10 +2x and the utility of choosing strategy B is U(B) = 10+3x. What is the Nash equilibrium to this game? Explain.